Code No: R05010102 Set No. 4

I B.Tech Regular Examinations, Apr/May 2007

MATHEMATICS-I

( Common to Civil Engineering, Electrical & Electronic Engineering,

Mechanical Engineering, Electronics & Communication Engineering,

Computer Science & Engineering, Chemical Engineering, Electronics &

Instrumentation Engineering, Bio-Medical Engineering, Information

Technology, Electronics & Control Engineering, Mechatronics, Computer

Science & Systems Engineering, Electronics & Telematics, Metallurgy &

Material Technology, Electronics & Computer Engineering, Production

Engineering, Aeronautical Engineering, Instrumentation & Control

Engineering and Automobile Engineering)

Time: 3 hours Max Marks: 80

Answer any FIVE Questions

All Questions carry equal marks

⋆ ⋆ ⋆ ⋆ ⋆

1. (a) Test the convergence of the series

p2−1

32−1 +

p3−1

42−1 +

p4−1

52−1 + ..... [5]

(b) Examine whether the following series is absolutely convergent or conditionally

convergent 1 − 1

3 ! + 1

5 ! − 1

7 ! + . . . . .. [5]

(c) Verify Rolle’s theorem for f(x) = log h x2+ab

x(a+b)i in [a,b] (x 6= 0). [6]

2. (a) Show that the functions u = x+y+z , v = x2+y2+z2-2xy-2zx-2yz and

w = x3+y3+z3-3xyz are functionally related. Find the relation between them.

(b) Find the centre of curvature at the point a

4 , a

4 of the curve √x +√y = √a.

Find also the equation of the circle of curvature at that point. [8+8]

3. (a) In the evolute of the parabola y2= 4ax, show that the length of the curve from

its cusp x = 2a to the point where it meets the parabola y2 = 4ax is 2a(3√3

- 1)

(b) Find the length of the arc of the curve y = log ex

−1

ex+1 from x = 1 to x = 2

[8+8]

4. (a) Form the differential equation by eliminating the arbitrary constant : log y/x

= cx. [3]

(b) Solve the differential equation: ( 1+ y2) dx = ( tan −1y – x ) dy. [7]

(c) The temperature of the body drops from 1000 C to 750C in ten minutes when

the surrounding air is at 200C temperature. What will be its temperature

after half an hour. When will the temperature be 250C. [6]

5. (a) Solve the differential equation: d3y

dx3 + 4dy

dx = Sin 2x.

(b) Solve the differential equation: x2 d2y

dx2 − 2x dy

dx − 4y = x4. [8+8]

6. (a) Find L [ t2 Sin2t ] [5]

(b) Find L−1 h s+3

(s2−10s+29)i [6]

1 of 2

Code No: R05010102 Set No. 4

(c) Evaluate

π/4

R0

a Sin

R0

r dr d

pa2 −r2 [5]

7. (a) Prove that ∇ × A¯×¯r

rn = (2−n)A¯

rn +

n(r¯.A¯)r¯

rn+2

(b) If ¯ F = (x2 − 27) i−6yzj +8xz2k evaluate RC

¯ F.d¯r from the point (0,0,0) to the

point (1,1,1) along the straight line from (0,0,0) to (1,0,1), (1,0,0) to (1,1,0)

and (1,1,0) to (1,1,1) [8+8]

8. Verify Stokes theorem f=x2i-yzj+k integrated around the square x=0, y=0, z=0,

x=1, y=1 and z=1.

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